Det Kongelige Danske Videnskabernes SelskabMatematisk-fysiske Meddelelser, bind 29, nr. 17

Dan. Mat. Fys. Medd. 29, no. 17 (1955)

FOURTH ORDER

VACUUM POLARIZATION

BY

G. KALLÉN AND A. SABRY

København 1955i kommission hos Ejnar Munksgaard

CONTENTSPage

I. Introduction 3II. General outline of the method 4

III. Discussion of the part .l1 (p2) 9IV. The real part of the vacuum polarization kernel 14

Appendix 17

Printed in DenmarkBianco Lunos Bogtrykkeri A-S

The real and imaginary parts of the kernel for the fourth order vacuum polar-ization are calculated for all values of the four-dimensional energy momentumvector. If an expansion in powers of the square of this quantity is used, thefirst coefficient agrees with a result previously obtained by BARANGER el. al.

I. Introduction.

I n a previous paper, one of us' has developed a formulationof renormalized quantum electrodynamics that is slightly

different from the standard techniques used by most authors.This modification was introduced because of its convenience indiscussions of general principles. It has been applied, for example,to a discussion, avoiding perturbation theory, of the magnitudeof the renormalization constants.' In the present paper, we wishto show that the new method can also be used with advantagein practical calculations in which perturbation theory is applied,and, as an illustration, the fourth order vacuum polarizationhas been chosen. BARANGER, DYSON and SALPETER 3 have com-puted those terms in this effect which are important in the Lambshift. They present, however, only the result and very few inter-mediary steps of the calculation. On the other hand, we attemptto give a fairly detailed account of our calculations, and computenot only the ternis of immediate experimental interest, but alsothe complete vacuum polarization kernel as a function of thefour-dimensional momentum. As will be seen later, our calculationis simplified to a certain extent by the fact that we can use theresult of an earlier calculation of the lowest-order radiativecorrections to the current operator 4 and thereby avoid some

I G. KALLEN, Hely. Phys. Acta 25, 417 (1952), in the following quoted as I.2 G. KALL)N, Dan. Mat. Fys. Medd. 27, no. 12 (1953).3 M. BARANGER, F. J. DYSON, and E. E. SALPETER, Phys. Rev. 88, 680 (1952).4 J. SCHWINGER, Phys. Rev. 76, 790 (1949).

1*

4 Nr. 17

integrations. Since the main work involved in the calculation ofa high-order effect is connected with the integrations over theso-called "Feynman auxiliary variables", a simplification at thispoint is not without interest. A further advantage of our methodis that the questions of regularization s and of the so-called "over-lapping divergences" 2 are completely avoided. Finally, due to theapplication of the known expression for the current operator, weneed not carry out any explicit mass renormalization in ourcalculations.

II. General Outline of the Method.

We start from the following formulae given in I:

<tt 7L

exi

O IJfn(x) I O\ t 4 ^Ipe t^'x (-17(p2)++T1(0)— in (p2))J^l(p), (1,

/(2 n)

17 (p 2) - 17(0) =- -p2 ^^da

^+ p2) )

(2)

Vn(p 2) = 3 9 ^^<01j,,Izi<z^.i1^10%• (3)

— p-^p

The notation is the same as in I and will be used here withoutfurther explanation. If the matrix elements of the current opera-tor are expanded in powers of e,

= J lel + 2J +e3 (2) + • 4e ^ e µ(f) Jµ • ( )

the first non-vanishing contribution to the function 17 (p 2) willbe

H(0)(p 2) _ _ - Ve2 2 ^ < ^ 1.lft l l '°^>< z lJµ l l Oi• (5)— 3 p pm = p

' W. PAULI and F. VILLARS, Rev. Mod. Phys. 21, 434 (1949). The regularizationof flic fourth order vacuum polarization has been discussed by E. KARLSON,Arkiv f. Fysik 7, 221 (1954).

2 A. SALAM, Phys. Rev. 82, 217 (1951). For the special problem of fourthorder vacuum polarization, the overlapping divergences have been discussed byR. Jost and J. M. LUTTINGER, Helv. Phys. Acta 23, 201 (1949).

4m 2 5 ' '^m22

—+^1 — ^p2 p-

e 212 ace

Nr. 17 5

This expression can be computed easily and gives

e 2 2m2)P^

H(0) (p2)12 7L2

(1 4 m2 1 + p$ 9(— p 2 -4 m 2), (6)

B (x) = 21 [1 +1x, (6a)

f( f)) (p2) — H(o) (0)

1+m2 4 2 -log

P

4 m21 + 2

4 m21 +

P2

The subsequent terni in the expansion of the function H (p 2) isof order e 4 , and contains the following terras:

II(t) (P 2) _ Ilål> (P 2) + .Mbt)

(P2),

Ve4 in) (P2) — p2 ^^O j ^

`) j^^^^zi<z

1," U , (9)3

417bt) (p 2) =

Ve 2 ^<0^)E^ lzi<z^j^ +^^^0^complexconjugate. (10)

--- 3 P p (=^ = P

The expansion of the current operator has been computed ear-lier. 1 From these results it can be seen that the term (9) getscontributions from states with one in-coming pair and one in-coming photon. 2 These matrix elements are

e2 1 ( ) (— l

)11YY (q +k)—my

< O1jµ > ^q ,q^,k> = v2

0)5 c F e

2 qk —,u2

_ y e Y

2qq, k k)^

!n Y^^] u(+> (q)J

The notation in the last expression is self-explanatory, exceptpossibly for the quantities u(±) (q). These are the normalized

' Cf., e. g., G. KÄLLN, Arkiv f. Fysik ", 371 (1950).z For the definition of particle numbers for these physical states, cf., e. g.,

G. KÄLLÉN, Physica 19, 850 (1953).

(8)

6 \r.1 7

plane-wave solutions of the free-particle Dirac equation. Theindex (+) refers to solutions with positive energy and the index(—) to solutions with negative energy. The vector e is the po-larization vector of the photon; V is the volume of periodicityand ,u a small photon mass introduced to handle infrared diver-gences. In the computation of the function H» (p 2), we must"square" the expression (11) and sum over all states wherek + q + q' = p. Using well-known properties of the functions u,and taking the limit V-te oc, we can write this sum as an integral

(2 n

.1^.Ie4 VZ< O ^ (pi) q,q',k><k,q',q ^)10>=4 +q' +k=p

e4

ss

dkdgdg' (p—g q'—k)b(q2+m2)0(g)b(q'2+m2)8(q')

xS (k2 ^-,u2) e (k) Sp [( z Y q' + m) (a)^ ZY(g +k) 21n Y72 gk—,u

iy (q'+k) +miy (q'+k) + mY7 ^ q , k ^^ 2 Yµ^ ( i Y q—ni) Yµ 2 q,k_p2 Y^

_ iy(q+k)—m ^)1•YR 2 qic —,u 2 Y J

The evaluation of this integral, which is the main task in ourcomputation, is given in a later paragraph.

The first approximation to the current, jO, has matrix ele-

ments which connect the vacuum only to states with one in-coming pair. Hence, the expression (10) will reduce to a sumover states with one in-coining pair

7 4l7bi) (p 2) -

tie ^ < q, q ' > < g', q1/013> + complex conjugate. (1^i

p- q+ q, = p

As has been mentioned in the introduction, the matrix elements

< 0 I j ) q, q' > have been computed by SCHWINGE R. 1 We write

his result as

e2 < 01

j µ'

1

q , q '> [—IZ<0>(p2) +iz(°'(0) + R(p2)-R(0) +S(0) —in 00'(p2) ^ (14

—R (p2))] <01 j`^' I q , q'> ^ (q —g 4 ) [S (p2) + i^ S (p 2)] <O 1w`°'^we"' 1 g, g'>,^ 2 m ^

Footnote 4, p. 3.

)

(12

I^m'-I

1+^p2 P2 + 4 m2 - 34 m2

log l — ^^) 9

l p1+

/ 4 m21 4----2-19(— p 2 -4 m 2), (15)e 2

«p2)8 :7

,/ 22 V1+4m

•log P

P21+ 1+4m

2

3+

1+ 2m 2

-}1+ 1-4m2

l/1 +^ ;210ß

4 z

V P

1i

(17)

Nr. 17 7

2`1 -1/1+4p2

e 2

I 1 + 2 212

(p2)— R(0) — 472 1

P

V1+

4m2 1+ l/1 +4m2/

P" P

,/ ° 4m21 1+V1+ 4p 1+1/1 ^

13-s-

4 log2

Vi 4 m2 ^ log 4 mz

1-- 4- 9 1—l/ 1+ p2P

^ 4m21+1/1

+4m2

1 + 2 log V p 2 21 / P 1

-1/1 +4m

P"

n2 j

4+ 9 (—P2/

(16)

_ e 2 m 2 Ø(—p2 -4 m2)S (p2)

4/ 2

2 P2 4 mV 1+ P

,^ 2 l + 1 + 4 _m2

S(p 2) = e2 ^ P _^ log ̂

P22 (18)

8ac'/1+4 m 1 - V

1{ 4 mP ^^//

P

The connection between the functions R (p2) and R (p2), and be-tween S (p2) and S (p2), is the same as the connection between11 (p2) and II (p2), which is given in Eq. (2). This is a conse-uence of the "causal" structure of the theory which says that

the value of the current in one point x can depend only on the

8 vr.17

previous history of the system inside the retarded light-cone be-longing to x. If Eq. (14) is written in x-space, we get a relationof the form

< 0 I jµ)(x)1q,q'i = ^ dx'F(x-- x')< 0I jµ)(x') I g,g'>

+G(x—x')<OI a 8xx') V(°'(x')--17`°'(x') a a (x^) I(J (1'> .1µ f^

Causality requires F (x) and G (x) to vanish if x0 < 0 and thisgives, in a well-known way, the relations involving the Hilberttransformations. This offers a new possibility of computing thematrix element under discussion by first computing the "imag-inary parts" R (p 2) and S (p 2), which can be obtained by inte-grating over finite domains in momentum space and, subse-quently, computing the "real parts" with the aid of Hilberttransformations. Actually, a calculation of this kind has beenperformed. However, it has not been found to be much simplerthan the standard methods for this problem. On the other hand,arranging the computation in this way is certainly not a morecomplicated procedure. We will not insist on this point here,but accept the results (14) — (18) as they stand. Consequently,the computation of the function 11 (b1) (p 2) will be reduced tosimple algebraic manipulations of these expressions. The functionØ (x) in (16) is defined by the integral

Ø (x) = log l 1 + t . (19)

Hereby it is supposed that the argument x is real, i. e. that4 m 2

1 + 2- > O. This will be sufficient at this stage. The integralp

Ø (x) has many interesting properties which will be of someuse in our calculation and that are discussed in the Appendix.

We now write the function H' (p 2) as

(14a)

17 b1) (p2) = 2 11" (p2) [—rl"(p2) + Iz<<,> (o ) + R (n2)— R (0 ) + S (0 )] + 2 S (p2) X (p2) ,

I(20)

Nr. 17

9

where

4e 2 X(p2) p2 (2

1703 dq dg S(p—q—q')(21)

X 6 (q2 + 111 2) 0 (q) 6 (q'2 + m2) 0 (q') (q'q' + ni z) . I

The last expression is easily computed

e 2 4 m2 s^s

X (p2) 247r2( 1 + 2 ) 0 (— p 2 — 4 m2). (22)

P

Collecting all these results, we have

^bt^(P2) 4 4[(3 - 5 2)(- 1 23 +S 2)+ a (7 L

4— 3 S +

3 S)log 1 ± S

II-1 1+52 1+5^ 1 1 S'

^- log S (3-52)^l— 2 5

logï-6

^2(3- 52)(1 +52)(Ø^- 1+ S )It

n2 1 1 + 5 1+ 5 1 +S')1

4 + 4 1og 2 1S — log l—S'log 9

5 0 (1 —5),

where

S = ^ / 1 +^-m 2 i0.Y P

III. Discussion of the Part 17Q1)(p2).

The remaining part of the function 11 (p 2), the integral (12),can be treated in the following way. We first compute the traceof the y-matrices. This is a straightforward calculation and thenecessary work can be considerably reduced by performing firstthe summations over the indices ,u and A. This can be done withthe aid of the well-known formulae

Ya. Yv, Yv, - • • • Yv,,, +, Y,, = 2 Yv= „+, . . . . Yv, Yv,

YA Yv, Yv, Y; = 4 Sv, v, .

(24)

(25)

(26)

(23)

10 Nr. 17

'l'hc complete trace can then be written

Sp [• • ÏS(i) (q, q')

S(i) (q ', q)

S(2) ( q , q') + S(2) ( q ', q)

( 2 qk— ,u2)2 + (2 q' k — ,u -' )2 + (2 qk—,u2) (2q' k —,u2),(27

S(1)(q, q ') = — 32 [kg • kg' + 2 m 2 qk + m 2 q'k + m2 (qq' — 2 rn 2)] , (2 F

S(2) ( q , q') = — 16 [2 ( gg') 2 — 4 m 2. qq' + 2 (kg + kg') qq' — m 2 (kg + kq')]. (29

Terms containing /2 2 have been dropped in (28) and (29), asthey will obviously vanish in the limit ,u 0.

Our next task is to compute an integral of the form

J = dk dg dg' b (p—k—q—q') S ( g2 + int) å ( q ' 2 + m 2) 6(k2 + ,u2)

(30

X 0 (q) O (q') O (k) F (qk, q'k, qq').

This can conveniently be done in two steps. We first consider

^ (p'2, Lp') _ dg a q2 + m2) 0 (q) b ((p ' q)2 + m2) 0 (p ' — q)

X F (gk, p'k — qk, p'q — q2).

This is a finite integral and we compute it in the special coordinatesystem where the space-like components of the vector p' vanish.

We then obtain

1(—pot, —kop,) =a x F(x,— kop',—x,in2- 1 p,2) 0(po--- 4 m2), (32d

2p, Iid 2^ x,

Ix1,2 = kopo + IkI1. 1 Dn 2 — m2. (32 a)

We now write this result in an invariant way, as

p (112 , kp1) — 2vucp)2 +p2p2 ^ dxF(x, kp' —x, m2 +-9p'2)B( p' 2 -4m 2), (33

4 mQ1/y2

+µ,2p2. (34 a)^

l p2—µ2 + 2 y

Nr. 17 11

2 2 kp + 2 V(kP )2+ 112p' 2 ' l 1

4 m-± ^2 ^ (33 a)

and treat the next integration similarly. The result is

J —. ,u. 8 (k2 + ,16 2) e (k) 1 ((p — k) 2 , kp + /4,2) =

S4-$-^ dy dzFl z— Z+µ2-z-2y +µ2, 9+m2+ p 2

2 u2),p- 1

(34)

Applying this technique to the integral (12), we get

171(11)(p2)

—4(p2 +4m 2) +E

e21 (A + B) B (— p 2 — 4 m 2),

12 7,c4 p4

A = dy dz l +m2— y +y 2 n1 2 )1 ,2 z—y y2 —

—1;

1t- É (p' - 1 ne') 7+ E

B = dy 2 2dz (p —2n1 )n12

+ 1 2+ 2 m2lp^ •(z y)2 2 y2 z2 J! Nt -P' —E

(36)

(37

To obtain these expressions, we have introduced the quantitiesy and z into (27), which becomes

i_ 1 2 y m2 (p2 2 m2)Sp^ ... - 32 2 f

z— y + (z —y)^

+ 1 z2 (y(p2 —m2) +^(p4- 4 n14))I

y2.

I (38)

The quantity A will stay finite in the limit µ — 0 and can be ex-pressed in elementary functions. After some straight forwardcalculations we get the result

12 Nr. 17

A = p4 12b (5 — 3 b Z) - (o + b2 — 3 b4) log +- . (39)64

The quantity B is a little more tricky to handle and the limit—k 0 cannot be performed in all terms. We write (37) as

B = g p4 (3 —b2)ti

^ — (1 62) Bo.) (1 ± 6 2) Bt2t], (4U )

where/3(1)t ^^ = 2 I (1), (41)

B t21 = S@) dz, I (42)

,— å a" a= -y2-62 .dy

=I (z) — 2 dy -y2 z2

1— 621— i --- -y (43)

o g _ 2f^ VI p` E y-- z

(y2 E)I1— _ rr 1

-52 1,

lllL 1 — ,y

2 ,u(43 a)E _ -- 2 •

The term containing the logarithmic dependence on ,z in I(z)can be split off in the following way:

I (Z) =

026,

dyy

1 —b21 — 1 -_

y b

1 — 6

2 ZZ. (44). (44)Vy2 —E2• b dy

2 — 1u2 ^2 (y2

y y E2)Yt E1 — b 2 1

1 Z2— 1 —\ 1— y

/I

2In the first integral, we make the transformation 1 —

82 = t`

and rewrite it as y

,a' V1-8'162d y b • V 1— 62 Y2 å dt y 1 —z262 (1 —e2/y2) 1 —z262 1-1

.1.dt 2 t 2 1

b 1 _ t [(1 + t) (1 — z2 62 12 ) 1 62 z2]

(45)

t 11

Nr. 17 13

The remaining integrations can now be performed withoutdifficulty, and I (z) is found to be

(z) 1 zlo 1+ b?

-lo 1+S

1+ 3 62z l0 1+

Sz

() 1z2( g l-Sz g1- 5/ - (2 z 2 (1 - S2 z2) g l—S-

S l0 8 52

+ 1-62 z2 g e(1-52)

The last integration in (42) introduces again the function 0(x).With the aid of the formulae given in the Appendix we can writethe result as

=- 3 Ø (_ 1-S ^ Ø 1 - 5 32+1or1+-

5 [log in1 1 + S ) - ^ 1+S 4 g1-S g((

'log 1+ S

log (1 + 5)2

+4 g +l -S g 4S

The remaining part of the calculation is purely algebraic innature. Collecting previous results, we get

47a(ß (p2)

e

48 7r4

3 -52 lo (

^( ) g 1-523

(3 52) (1+ 52)

5 (39 17 6 2 ) + i8 16

64 S4 )

S (3 6 2 )(

)

(33

I 1-

Ø 1S

- 10 6 2 + 6 4) 1og

1+ 62 log

j

2 S g 1 -S

3 n 2_1+ 5 ) + 4 4g

1 ±1 -

g • lo f̂^

lo

21+ 5

1 - S

(48)6) 2 3 Ø e1 + 5 + ^12

-- 4)( 1og 1 + b log ( 1 -1-

6)2/1 (1 - S).1- S 4S

Adding (48) and (23), we find that the terms depending on ,u

cancel. In this way we obtain

7(1)(132) a2 -19+5562-54_.3 62 log 6454

- 3 7c 2 I 24 7 9 3 2

log (1-62A ,

1+- S [33 23 23 1 354^

(1 + (5)31^log1 -S 16 + 8

S 2 1654^--6Ss+^-9+52- 9

log - 8 b2

- (+ 52 2 +54

) [4 Ø^, 1S + 2 Ø( 1 + ^ ^3^0(1 - S),2

(46)

(47)

(49)

14

e 2

4 z

Nr. 17

(49 a)

This is our expression for the imaginary part of the kernel (1).According to (2), the real part is obtained after a Hilbert trans-formation of this expression. This kill be discussed in the nextparagraph.

IV. The Real Part of the Vacuum Polarization Kernel.

So far, all our results are given as functions of the quantity 6defined in (24). It is therefore convenient to introduce a newvariable of integration instead of a in (2). If we put

4 m2 = z 2a

we get

11(»(p2)—T7(1)(0) = 2zdz

62

Ht

r t (6 = z) .(51)

Not all the integrations in (51) can be carried out explicitly, withthe result expressed by elementary functions or by the function

(x). The new integrals which appear can be written in thestandard form

F(x,y)= •t

logI 1 + xt I . log Il + yt I. (52)

0

All the necessary integrals over the function Ø (x) can be ex-pressed in terms of this F(x, y)

(50)

0

,«dz

P z +b Ø(z) = Ø (a) log

0

1a

+ b— F (a, b). (53)

In our final result, one of the variables in F (x, y) has only avery small number (three) of different values. We thereforeintroduce the following three integrals, each of which dependson only one variable:

Nr. 1 7 15

.+ I

F (x) = ft log (1 -}- t) log 1— t^x

(54)

•+1dt

1 tG (x) log • logt2

1 — -- (55)1+ t 2 x

I I.

rdt

log I t1 1ogH (x) i

1 ^x

(56)

—1

The remaining integrations are then straightforward, and nottoo time consuming. The result can be written as

1-1(1) 17^1) 0 _ a 213 + 11 62

1 b4 + b (19 — 55 a s(P2) — ( ) 3 n2 1 108 72 3 24 72

1 d4 I lo^ 1^- ô (33 + 23 62 23 64 + bs

°lo^r2 1 +

b n20 (1 b))-

3 / b ^1— b ^ 1 . 32 16 32 121 ^1— S^

+2

(1 } b +1 6))+ 2Ø( 1 ô

4 4n20(1—b) (57)

1

— 3 1og 2 1+ b } 1 log 1+ b log 64 64

4 I1— S I 2 I 1— b I 11—b213

+ (3 + 2 6 2 — 6 4) [F (62) + G (6 2) — H (62)

1 1

If this expression is expanded in powers of 6 -1, the first non-vanishing term will be of order 6-2 . The same conclusion canalso be obtained from a study of Eq. (51). If this expression isexpanded in powers of b-1 , we get immediately

+ LH(1) (p2) — 1/ (1)tt) (0) = 2 - Ç z dz H(1) (b = z) + .. • . (58)

62 .._1

The numerical coefficient of the first power of b-2 has beencomputed from Eq. (57) and with the aid of the integrationindicated in Eq. (58). The agreement of the results serves asa check on the calculations. In either way we obtain

16 Nr. 17

n(u(p2) HO) (0)l a 2 82 p2 a 2 41

bz ^ 2 81 + • • = — (59)• •

n-1 2 n2162

This also agrees with the result obtained by BARANGER, DYSON,and SALPETER.1

In Eq. (57) it is supposed that b is real, that is, p 2 is eitherpositive or less than — 4 nt 2. For 0 < — p 2 < 4 m 2, b will bepurely imaginary. In this case we have to substitute arctangentfunctions for logarithms, according to the following rules:

b log 1 ^ -^ 6 1 -4- 27] arctang -1, ( )) = i b > 0), ( 60 a)

log 2 ^ 1+ —^c 2 0 (1— b) -^^ -4arc#ang 2 ^, (60 b)

b + 2 Ø(- 1 - b ^-^2 - 3 n2 0(1 — b) - 3 1og 2 1 + bb Ø

(1 { (5) 1-^ b ) 4 4 4 11—b^

21og I i±

—61 log 64 d4

2I31 y^I2 a retang — 2 lp (2 arctang o (60 e

i4arctg 1 log 64 rn (1 +n2)3112)'j'

(x) _ V^ sinnnx)

(60 d)

At the point p 2 = — 4 m 2, or = 0, the expression (57) has alogarithmic singularity. If, (luring the calculation, the photon massu had been kept different from zero in all places, our result wouldhave shown a finite peak at this point. For practical applications,the weak logarithmic infinity will not be very harmful, as one isin general interested in convolution integrals involving the functionII(p2)-17(0). In such expressions, the result (57) will be suf-ficient. For large values of I p2 /m 2 I, our function behaves aslog e I p2/m 2 I. Fig. 1 gives a qualitative idea of the behaviourof the fourth approximation of the vacuum polarization kernelas a function of —p 2/m 2. A figure of the corresponding behaviourof the functions 17 (°) (p 2) and 17t°)(p2)-17(0)(0) would be rather

Footnote 3, page 3.

Nr.17 17

Fig. 1. Qualitative behaviour of the e 4 approximation of the real and of the imag-inary parts of the vacuum polarization kernel.

similar to Fig. 1. The only qualitative difference would be thatthe function II(° ) (p 2) vanishes at the point —p 2 = 4 m 2 and that11w function II(0) (p2) — II (° ) (0) has a finite peak at Ibis point_

Appendix.

In the following are given some formulae involving thefunction Ø(x), defined in Eq. (19). Although practically allthese expressions can be found in the literature,' we add this-summary for the reader's convenience.

If x is real, our function is defined by

If we consider

Ø (x) = S —dz log 11 + z I.

z

1 /xØ(x) =

^ —dz log l l +zl,

(A. 1)

(A. 2)

I Cf., e.g., K. MITCHELL, Phil. Mag. 40, 351 (1949) and W. GRÖBNER, N. Hor• -REITER, Integraltafeln, Wien and Innsbruck, 1950.

Dan. Mat. rys. Medd. 29, no 17. 2

nx

Ø (x)=lloglzllogll+zl 1+^loglzl.J i

1+ x

= loglxl •logl 1 +xl— z

logl 1 — zl,d

(A. 5)

■■- 2

18 Nr. 17

and make the variable transformation t = z `, we get the funda-mental relation

1 +.+i

?B (—x) dz logl

i +4lA. 3)

_ —Ø (x)+2–logtilx^2

^ B(— x)

or2

Ø (x) -{- Ø (-1)

= -1 log e I x I — 2 Ø (— x) . (A. 4)

An integration by parts in the definition (A. 1) will give anotheruseful formula

or2

Ø(x)^-Ø( 1—.x) = — 3 + log l xl• log ll+xl. (A.6)

Besides (A. 4) and (A. 6), we also mention the formula

..r +1

0(x)±0(— dz log ll--z2 l+

dzlogll— z

(A. 7)2

= ^ 8Ø(— x2)— .

Another relation which has been of some use in the calculationscan be obtained in the following way:

Ç 1og

x

Ø (x) Ø (x) 1 + z ?ç2

-. (A. 8)z 1— z 4i

2 _Ø (x)—Ø(—.x)= 2 1

ØI

Ø(x)

- -1+ogI x I-log1 +

1 + x1—x

(A. 10)

Nr.17 19

The transformation 1 + t — 1 Z transfers this integral to+

2x

1+x r ll

Ø(x) — ( L) 4 — dt L f 2

+ t l logll+tl S

1

L 11

s0(-1 x) t(l t̂ t) logl 1+t^ = 4 —^2e(-1— x)

—Ø ( 1 + x/ + Ø (1 2+ x/.

Using (A. 6), we can write (A. 9) as

_ 07

For complex values of x we can still define the function Ø (x)as the integral (A. 1), making this definition unique with the aidof a cut along the real axis below the point —1. This functionfulfils an equation similar to (A. 4),

(x) + Ø l l = log' x, (A. 11)x 2

where the definition of the logarithm is made unique by theprescription just mentioned. From (A. 11), we conclude that

ReØ(e`^) = — 4̂ 2.

For I x I < 1, we have the power series expansion

n2 ) I(— 1)n + 1 ,xnØ (x) _

— 12 + _^ \ n2

(A. 12)

—

(A. 13)

20 Nr. 17

From (A. 13), it follows that

i^ ^sin(n a^ImØ(— e ) _ — ^u n2

—v(i9). (A. 14)

Numerical values of Ø(x) for real x can be obtained from thepaper by MITCHELL. The function tV (t) in (A. 14) has beentabulated by CLAUSEN1.

T. CLAUSEN, Jour. f. Math. (Crone) 8, 298 (1832).

GERN (European Organization for Nuclear Research),Theoretical Study Division, Copenhagen, Denmark

andInstitute for Theoretical Physics, University of Copenhagen.

Indleveret til selskabet den 18. februar 1955.F:erdig fra trykkeriet den 20. august 1955.